20060211, 00:36  #34  
Aug 2002
Buenos Aires, Argentina
2^{3}·13^{2} Posts 
Quote:


20060211, 06:33  #35  
Nov 2003
2^{2}×5×373 Posts 
Quote:
quadratic reciprocity which is not a high school topic. And the proof is a meaningless jumble of algebraic manipulations that does not convey *understanding*. The proof I have in mind gives an immediate "Aha! Of course!" to someone with a little knowledge of algebra. It shows an element of maximal order in a group of order P+1. And it is a lot shorter. 

20060211, 11:27  #36  
Jun 2005
Near Beetlegeuse
2^{2}·97 Posts 
Quote:


20060211, 17:22  #37 
"Phil"
Sep 2002
Tracktown, U.S.A.
1119_{10} Posts 
Actually, Dario made some excellent changes to the wiki that clarify the overall structure of the proof, but in the process, some of the constructions used in the first part do not get introduced until the second part. I plan to take those constructions out and place them before both the necessity and sufficiency proofs, so that those two proofs may be read in either order.
As for the suggestion of Bob Silverman, I think it would also be a great addition to this section of the Wiki. But there is still a role for this proof which seems to require relatively little in the way of prerequisites. By the way, I thought the explanation of quadratic reciprocity on the wiki was wellwritten and should be helpful to anyone needing help understanding the LL test proof. 
20060211, 23:10  #38  
Jun 2005
Near Beetlegeuse
2^{2}×97 Posts 
Quote:


20060213, 19:02  #39 
Aug 2002
Buenos Aires, Argentina
2^{3}×13^{2} Posts 
I think that in the Proof of Necessity of LucasLehmer test MersenneWiki article, we would have to first state that 3 is not a quadratic residue modulo Q and then start working with numbers of the form , in order to show that the number is being added to the field of numbers modulo Q, like we add the number to the integers to generate the Gaussian Integers. What do you think?

20060213, 19:19  #40 
Aug 2002
Buenos Aires, Argentina
2^{3}·13^{2} Posts 
Suppose that we need to prove that does not have solutions for positive integers x, y, z. What proof do you like?
1) Since Wiles' proof of the Last Theorem shows that has no solutions for positive integers x, y, z when n>2, then when n=4, the proof follows. 2) Use original Fermat proof: Proposition: There are no integer solutions of . PROOF: Suppose there are integers x,y,z such that . This can be written as a Pythagorean triple , from which it follows that , , and . Since is a square, we know that either or is even. Thus, from the Pythagorean triple we have , , and . Also, since is a square we can set and . Now, since , we have and . These, along with , can be substituted back into to give , where v is smaller than z, contradicting the fact that there must be a smallest solution. Notice that the first proof is only two lines long, and the second one is about 15 lines long. But the first one requires Wiles' proof that is about 100 pages long. When we include the proof of all theorems needed in its demonstration recursively (so it can be followed by someone that has only highschool math education) we will need probably about 10000 pages (this is a wild guess). In the second proof we would have 15 lines plus other 15 in order to show the form of Pythagorean triples. Well, what demonstration do you finally prefer? This is the same case when we use highlevel languages in computing science. A 3line source code can generate a 1 MB executable file while a 1000line source code can generate a 10 KB executable file, because the first one uses a function that needs a very large library. 
20060213, 21:30  #41  
Jun 2005
Near Beetlegeuse
2^{2}·97 Posts 
Quote:
Let's suppose that . Then you say that: Quote:
Now make . This gives , and . And yet you then go on to claim that where, I'm afraid, the unavoidable impression is that we are no longer looking at an integer solution of anything. At which point I gave up. This is the sort of proof that definitely does not give that "Aha!" moment that Mr Silverman was talking about. It just leaves me frustrated that even after studying 16 hours a week I still cannot make head or tail of what you are talking about. 

20060213, 23:10  #42 
Jun 2005
Near Beetlegeuse
604_{8} Posts 
It is of course perfectly possible that the above says more about my ability to study than it says about your ability to write proofs.

20060214, 22:21  #43  
Aug 2002
Buenos Aires, Argentina
2^{3}·13^{2} Posts 
Quote:
Quote:
Quote:


20060215, 01:27  #44  
Bronze Medalist
Jan 2004
Mumbai,India
804_{16} Posts 
Quote:
http://homepages.cwi.nl/~dik/mathematics/jsh2.html Mally 

Thread Tools  
Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Proof of Fermat's Last Theorem  McPogor  Miscellaneous Math  18  20071019 11:40 
help with a proof  vtai  Math  12  20070628 15:34 
Proof (?!) that RH is false?  bdodson  Lounge  6  20070319 17:19 
A proof with a hole in it?  mfgoode  Puzzles  9  20060927 16:37 
A Second Proof of FLT?  jinydu  Math  5  20050521 16:52 